Question

Barbara makes a sequence of 28 semiannual deposits of the form X,2X,X,2X,…X,2X,X,2X,… into an account paying a rate of 6.8 percent compounded annually. If the account balance 8 years after the last deposit is $9100, what is XX?

Answer 1

Here Compounding will be for separate Annuity one for X and another for 2X.

X payment started from 0.5 Years. and it will be for 0.5,1.5,2.5 ..to till 7.5 Years. (Total 08 Nos of Payment till 7.5 Years)

Now Annual Rate of Interest r = 6.8% = 0.68

So Value of X Annuity after 7.5 Years

.

Applyingformula Sum of Geometric Series

= X * 10.186

Now at the End of 08 Years this Value becomes V1

Semi annual Interest Rate = Annual Rate / 02 = 6.8% / 2 = 3.4% = 0.034

V1 = Amount * (1+ Interest Rate) = X * 10.186 * ( 1+ 3.4%) =   X * 10.186 * 1.034 = X * 10.532

2X payment started from 1 Year . and it will be for 1,2,3 to till 8 Years. (Total 08 Nos of Payment till 08 Years)

So Value of 2X Annuity after 8 Years V2

= 2X * 10.186

So total Balance at end of 08 Years become

V1 + V2 =  X * 10.532 +  2X * 10.186 = X * 30.904

Now Actual Balance is 9100

X * 30.904 = 9100

So X becomes =  9100 / 30.904 = 294.45

Value of 2X = 2* 294.45 = 588.90 (Ans)

Ans :2X Value = 588.90